Theorem cnvss 4869

Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
cnvss	⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Proof of Theorem cnvss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3195	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2		df-br 4060	. . . 4 ⊢ (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3		df-br 4060	. . . 4 ⊢ (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
4	1, 2, 3	3imtr4g 205	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
5	4	ssopab2dv 4343	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6		df-cnv 4701	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7		df-cnv 4701	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
8	5, 6, 7	3sstr4g 3244	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ∈ wcel 2178   ⊆ wss 3174  ⟨cop 3646   class class class wbr 4059  {copab 4120  ^◡ccnv 4692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-br 4060  df-opab 4122  df-cnv 4701

This theorem is referenced by:  cnveq  4870  rnss  4927  relcnvtr  5221  funss  5309  funcnvuni  5362  funres11  5365  funcnvres  5366  foimacnv  5562  tposss  6355  structcnvcnv  12963  pw1nct  16142